Recent zbMATH articles in MSC 46Fhttps://zbmath.org/atom/cc/46F2024-03-13T18:33:02.981707ZWerkzeugVanishing of certain equivariant distributions on spherical spaces for quasi-split groupshttps://zbmath.org/1528.220082024-03-13T18:33:02.981707Z"Lu, Hengfei"https://zbmath.org/authors/?q=ai:lu.hengfeiThis paper contains an important result concerning the vanishing of some eigendistributions on a quasi-split real reductive group. Let \(G\) be a quasi-split real reductive group, \(B\) be a standard Borel subgroup of \(G\), \(U\) be the corresponding unipotent radical of \(B\), and let \(\Psi\) be a non-degenerate character of \(U\). Let \(H\) be a spherical subgroup of \(G\), and let \(\chi\) be a character of \(H\). Let \(Z\) be the complement to the union of open \(B\times H\)-double cosets in \(G\), and let \(\mathfrak{z}\) be the center of the enveloping algebra \(\mathscr{U}(\mathfrak{g})\) of the Lie algebra \(\mathfrak{g}\) of \(G\). The author proves (Theorem 1.1) that there are no non-zero \(\mathfrak{z}\)-eigen \((U\times H\), \(\psi \times \chi\))-equivariant distributions supported on \(Z\).
Theorem 1.1 is a generalization of a theorem from [\textit{A. Aizenbud} and \textit{D. Gourevitch}, Math. Z. 279, No. 3--4, 745--751 (2015; Zbl 1315.22014)], where the authors obtained a similar result for a split real reductive group \(G\).
Reviewer: Allan Merino (Boston)Approximation of functions on rays in \(\mathbb{R}^n\) by solutions to convolution equationshttps://zbmath.org/1528.300132024-03-13T18:33:02.981707Z"Volchkov, V. V."https://zbmath.org/authors/?q=ai:volchkov.valerii-vladimirovich"Volchkov, Vit. V."https://zbmath.org/authors/?q=ai:volchkov.vitalii-vladimirovichSummary: This is a first study of approximation of continuous functions on rays in \(\mathbb{R}^n\) by smooth solutions to a multidimensional convolution equation with a radial convolutor. We obtain an analog of the well-known Carleman's Theorem on tangent approximation by entire functions. As consequences, we give some new results of interest for the theory of convolution equations. These results concern the density in \(\mathbb{C}\) of the range of some solutions to the convolution equation as well as the possible growth of solutions on rays in \(\mathbb{R}^n \).Kernel theorems for Beurling-Björck type spaceshttps://zbmath.org/1528.460022024-03-13T18:33:02.981707Z"Neyt, Lenny"https://zbmath.org/authors/?q=ai:neyt.lenny"Vindas, Jasson"https://zbmath.org/authors/?q=ai:vindas.jassonThe authors consider a general class of Beurling-Björck spaces (of Beurling and Roumieu type) defined in terms of two weight function systems \({\mathcal V}\) and \({\mathcal W}\). In the case that \({\mathcal V} = \{e^{\frac{1}{\lambda}v}: \lambda > 0\}\) and \({\mathcal W} = \{e^{\frac{1}{\lambda}w}: \lambda > 0\}\) for appropriate non-negative continuous functions \(v, w\) on \({\mathbb R}^d\), the classical Beurling-Björck spaces are recovered. Section~3 (along with Appendix~A) characterizes when the general Beurling-Björck spaces are nuclear in terms of the weight systems, while Section~4 is devoted to the new kernel theorems. An important tool in the proof of the kernel theorem for Roumieu-type spaces is the projective description of said spaces using maximal Nachbin families associated with weight systems.
Reviewer: Antonio Galbis (València)Finite decomposition of Herz-type Hardy spaces for the Dunkl operatorhttps://zbmath.org/1528.460292024-03-13T18:33:02.981707Z"Lachiheb, Mehdi"https://zbmath.org/authors/?q=ai:lachiheb.mehdiSummary: The corresponding Herz-type Hardy spaces to new weighted Herz spaces \(HK^{\beta, p}_{\alpha, q}\) associated with the Dunkl operator on \(\mathbb{R}\) have been characterized by atomic decompositions. Later a new characterization of \(HK^{\beta, p}_{\alpha, q}\) on the real line is introduced. This helped us in the work to characterize that the norms of the Herz-type Hardy spaces for the Dunkl operator can be achieved by finite central atomic decomposition in some dense subspaces of them. Secondly, as an application we prove that a sublinear operator satisfying many conditions can be uniquely extended to a bounded operator in the Herz-type Hardy spaces for the Dunkl operator.The position-momentum commutator as a generalized function: resolution of the apparent discrepancy between continuous and discrete baseshttps://zbmath.org/1528.811702024-03-13T18:33:02.981707Z"Boykin, Timothy B."https://zbmath.org/authors/?q=ai:boykin.timothy-bSummary: It has been known for many years that the matrix representation of the one-dimensional position-momentum commutator calculated with the position and momentum matrices in a finite basis is not proportional to the diagonal matrix, contrary to what one expects from the continuous-space commutator. This discrepancy has correctly been ascribed to the incompleteness of any finite basis, but without the details of exactly why this happens. Understanding why the discrepancy occurs requires calculating the position, momentum, and commutator matrix elements in the continuous position basis, in which all are generalized functions. The reason for the discrepancy is revealed by replacing the generalized functions with sequences approaching them as their parameter approaches zero. Besides explaining the discrepancy in the discrete and continuous models, this investigation finds an unusual double-peaked sequence for the Dirac delta function.